In the late 1960s, I was able to regularize and exponentiate Dashen and Sharp's algebra to obtain an infinite-dimensional group, and establish a framework for studying its unitary representations [1]}. The group is the natural semidirect product of a diffeomorphism group Diff\(_0(\mathbb {R}^3)\)  of the physical space (describing flows generated by momentum density operators), with an additive group of scalar functions on \(M\)  (describing exponentiated mass density operators). Then in collaboration with Grodnik, Powers, and Menikoff [2]}, [3]}, [4]}, [5]}, [6]}, Sharp and I found applications confirming the fundamental role of this group and its Lie algebra. Across the 1970s, Menikoff, Sharp, and I successfully extended Mackey's method of induced representations of locally compact Lie groups, to obtain a class of unitary representations of diffeomorphism groups [7]}. It followed from our work that Bose and Fermi exchange statistics could be understood as inequivalent unitary representations of our semidirect group, induced by representations of \(S_N\) . And \(S_N\)  entered naturally as the fundamental group of \(N\) -particle configuration space. This provided a new, wholly kinematical perspective on the earlier work of Laidlaw and DeWitt [8]}, which had been based on Feynmann paths; and it led to intermediate quantum statistics in two-space.
